In the U.S. certificate would not be understood when requesting a student's records. To get a record of courses taken and/or scholastic awards achieved one would be requesting a transcript, or one would simply request the student's records. A certificate would be the document one receives at graduation, attesting that the student did indeed graduate. Transcripts, to my knowledge, are not certified either, although you would get official transcripts, meaning there is some level of attested accuracy. Graduation certificates (diplomas) would be signed by the appropriate authorities.
As for the use of registration, in the US that is simply the act of signing up for the school or for classes. It would never refer to scholastic or academic records. If one wanted the current course schedule, then one could request a list of courses the student is registered* in. Once the courses have been completed, and exams taken, then the student will no longer be registered in those courses, and the courses will become part of the student's academic records. In the US, register as a list (noun) would not be used for academic records.**
Lastly, this is all hypothetical. I am reasonably certain that, if a 3rd party did request a student's records in the US, in any sense discussed here, they would be turned down flat, as it would be an invasion of privacy. They could find out if the student graduated, and what their major area of study was, but I think that would be about it. Of course, a student may request their own records.
*A comment has been made that the use of registration here is incorrect, and enroll would be preferred. In my experience this is not a distinction that is commonly made. This distinction may or may not be correct, I could not say. However, in my experience the two are nearly interchangeable, and this answer is regarding the usage of register/registration specifically.
**As a side-note for possible discussion, notice the differences in definition of register provided by Google (from Oxford Dictionaries) and that provided by the Cambridge Dictionary. The definition provided by the Oxford Dictionaries might be broadly taken to be usable for student records, whereas that provided by the Cambridge Dictionary does not seem to extend that broadly.